Abstract 1 Weber

In the first of the three lectures our understanding of homotopy 3 types will be reviewed. There is a symmetric monoidal closed structure on the category 2-Cat, of 2-categories and 2-functors, called the Gray tensor product. A Gray category is a category enriched in 2-Cat via this tensor product and a Gray groupoid is a Gray category in which the arrows, two cells and three cells are strictly invertible. Proofs have been given by Joyal-Tierney, Leroy, Berger and Lack that Gray groupoids provide an algebraic model for homotopy 3-types. In Lack's work, a Quillen model structure is exhibited on Gray-Cat for which the weak equivalences are the triequivalences between Gray categories. Taken together the facts recalled here could be labelled as a "direct algebraic understanding of homotopy 3 types". This gives an indication of how the general case for arbitrary n should go.

Gray categories are weaker than strict 3-categories, which is what one gets by enriching 2-Cat with the cartesian tensor product, but a lot simpler and stricter than tricategories in the sense of Gordon-Power-Street. The problem of identifying what the appropriate analogue of semi-strict-n-category is open. There has however been quite a lot of progress on the algebraic side of the story.

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